Hydrodynamics and Flow Tetsufumi Hirano Department of Physics The University of Tokyo The University of Tokyo QGP Winter School 2008 February 1-3, 2008, Jaipur, India QGP Winter School 2008 February 1-3, 2008, Jaipur, India

My Charge and Disclaimer To lead students/young post-docs to frontier of hydrodynamic description in H.I.C. in order for them to understand talks/posters to be presented in the upcoming QM2008 conference. Not to cover recent studies to be presented at QM2008.

Plan of Lecture PART 1 (INTRODUCTION) –Introduction to hydrodynamics in H.I.C. PART 2 (FUNDAMENTALS) –Formalism of relativistic ideal/viscous hydrodynamics PART 3 (APPLICATIONS) –Basic Checks –Elliptic flow –Ideal hydrodynamic model –Application of ideal hydrodynamic model to H.I.C. and comparison with data

PART 1 Introduction to hydrodynamics in relativistic heavy ion collisions

Why Hydrodynamics? Static Quark gluon plasma under equilibrium Equation of states Transport coefficients etc Dynamics Expansion, Flow Space-time evolution of thermodynamic variables Energy-momentum: Conserved number: Local thermalization Equation of states

Freezeout “Re-confinement” Expansion, cooling Thermalization First contact (two bunches of gluons) Longitudinal Expansion in Heavy Ion Collisions

Complexity Non-linear interactions of gluons Strong coupling Dynamical many body system Color confinement Inputs to phenomenology (lattice QCD) Bottom-Up Approach to Heavy Ion Physics The first principle (QuantumChromo Dynamics) Experimental Relativistic Heavy Ion Collider ~200 papers from 4 collaborations since 2000 Phenomenology (hydrodynamics)

Connection among Lectures Hydrodynamics QGP fluids Color Glass Condensate (Venugopalan) Jet quenching (Kharzeev,d’Enterria) J/psi suppression (Satz, Lourenco) EM probe (Srivastava) Particle production (Stock) Collective flow (Hirano)

PART 2 (FUNDAMENTALS) Formalism of relativistic ideal/viscous hydrodynamics

Relativistic Hydrodynamics Energy-momentum conservation Current conservation Energy-Momentum tensor The i-th conserved current In H.I.C., N i  = N B  (net baryon current) Equations of motion in relativistic hydrodynamics

Tensor/Vector Decomposition Tensor decomposition with a given time-like and normalized four-vector u  where,

“ Projection ” Tensor/Vector u  is perpendicular to  . u  is local four flow velocity. More precise meaning will be given later. Naively speaking, u  (  ) picks up time- (space-)like component(s). Local rest frame (LRF):

Decomposition of T  :Energy density :(Hydrostatic+bulk) pressure P = P s +  :Energy (Heat) current :Shear stress tensor : Symmetric, traceless and transverse to u  & u

Decomposition of N  :charge density :charge current Q. Count the number of unknowns in the above decomposition and confirm that it is 10(T  )+4k(N i  ). Here k is the number of independent currents. Note: If you consider u  as independent variables, you need additional constraint for them. If you also consider P s as an independent variable, you need the equation of state P s =P s (e,n).

Ideal and Dissipative Parts Energy Momentum tensor Charge current Ideal part Dissipative part

Meaning of u  u  is four-velocity of “flow”. What kind of flow? Two major definitions of flow are 2. Flow of conserved charge (Eckart) 1. Flow of energy (Landau)

Meaning of u  (contd.) Landau (W  =0, u L  V  =0) Eckart (V  =0,u E  W  =0) uLuL VV uEuE WW Just a choice of local reference frame. Landau frame might be relevant in H.I.C.

Entropy Conservation in Ideal Hydrodynamics Neglect “dissipative part” of energy momentum tensor to obtain “ideal hydrodynamics”. Q. Derive the above equation. Therefore,

Entropy Current Assumption (1 st order theory): Non-equilibrium entropy current vector has linear dissipative term(s) constructed from (V , ,  , u  ). We have assumed N  = 0, so the second term does not appear in this case. In what follows, we consider the Landau frame only and omit subscript “L”. For simplicity, we further assume there is no charge in the system.

The 2 nd Law of Thermodynamics The 2 nd thermodynamic law tells us Q. Check the above calculation.

Constitutive Equations Thermodynamic force Transport coefficient “ Current ” tensorshear scalarbulk

Equation of Motion : Expansion scalar (Divergence) : Lagrange derivative

Equation of Motion Q. Derive the above equations of motion from energy-momentum conservation.

Intuitive Interpretation of EoM Work done by pressure Production of entropy Change of volume Dilution Compression

Lessons from (Non-Relativistic) Navier-Stokes Equation Assuming incompressible fluids such that, Navier-Stokes eq. becomes Final flow velocity comes from interplay between these two effects. Source of flow (pressure gradient) Diffusion of flow (Kinematic viscosity, /, plays a role of diffusion constant.)

Generation of Flow x P Expand Pressure gradient Source of flow  Flow phenomena are important in H.I.C to understand EOS

Diffusion of Flow Heat equation (: heat conductivity ~diffusion constant) For illustrative purpose, one discretizes the equation in (2+1)D space:

Diffusion ~ Smoothing R.H.S. of descretized heat/diffusion eq. i j i j x x y y subtract Suppose  i,j is larger (smaller) than an average value around the site, R.H.S. becomes negative (positive). 2 nd derivative w.r.t. coordinates  Smoothing

Shear Viscosity Reduces Flow Difference Shear flow (gradient of flow) Smoothing of flow Next time step

Bjorken ’ s Equation in the 1 st Order Theory (Bjorken’s solution) = (1D Hubble flow) Q. Derive the above equation.

Viscous Correction Correction from shear viscosity (in compressible fluids) Correction from bulk viscosity  If these corrections vanish, the above equation reduces to the famous Bjorken equation. Expansion scalar = theta = 1/tau in scaling solution

1. and are dimensionless quantities in natural units and intrinsic properties of fluids. 2. One often says, “viscosity is small”. However, one has to say this in the context of heavy ion collisions in a more precise way: “Viscous coefficients are small in comparison with entropy density.” A Few Remarks

is obtained from SUSY Yang-Mills theory. is obtained from lattice. Bulk viscosity has a prominent peak around T c. Recent Topics on Transport Coefficients Need microscopic theory (e.g., Boltzmann eq.) to obtain transport coefficients. Kovtun,Son,Starinet,… Nakamura,Sakai,… Karzeev,Tuchin,Karsch,…

Necessity of Relaxation Time Non-relativistic case (Cattaneo(1948)) Fourier’s law  : “relaxation time” Parabolic equation (heat equation) ACAUSAL! Finite  Hyperbolic equation (telegraph equation) Balance eq.: Constitutive eq.:

Entropy Current (2 nd ) Assumption (2 nd order theory): Non-equilibrium entropy current vector has linear + quadratic dissipative term(s) constructed from (V , ,  , u  ).

Constitutive Equations Relaxation terms appear (  and   are relaxation time). No longer algebraic equations! Dissipative currents become dynamical quantities like thermodynamic variables. Employed in recent viscous fluid simulations.

Bjorken ’ s Equation in the 2 nd Order Theory New terms (written in red) appear in the 2 nd order theory.  Coupled equations where

Summary (part 1&2) Hydrodynamics provides a dynamical framework in heavy ion collisions towards understanding bulk and transport properties of the QGP. Na ï ve extension of Navier-Stokes equation to its relativistic version has a problem on causality.  Need the 2 nd order corrections in entropy current.

Contents PART 1 (INTRODUCTION) –Introduction to hydrodynamics in H.I.C. PART 2 (FUNDAMENTALS) –Formalism of relativistic ideal/viscous hydrodynamics PART 3 (APPLICATIONS) –Basic Checks –Elliptic flow –Ideal hydrodynamic model –Application of ideal hydrodynamic model to H.I.C. and comparison with data

PART 3 (APPLICATIONS) Basic Checks

Sufficient Energy Density? Bjorken energy density : proper time y: rapidity R: effective transverse radius m T : transverse mass Bjorken(’83) total energy (observables)

Critical Energy Density from Lattice Note that recent results seem to be T c ~190MeV. Adopted from Karsch(PANIC05)

Centrality Dependence of Energy Density PHENIX(’05)  c from lattice Well above  c from lattice in central collision at RHIC, if assuming =1fm/c.

Caveats (I) Just a necessary condition in the sense that temperature (or pressure) is not measured. How to estimate tau? If the system is thermalized, the actual energy density is larger due to pdV work. Boost invariant? Averaged over transverse area. Effect of thickness? How to estimate area? Gyulassy, Matsui(’84) Ruuskanen(’84)

Matter in (Chemical) Equilibrium? Two fitting parameters: T ch,  B direct Resonance decay

Amazing Fit! T=177MeV,  B = 29 MeV Close to T c from lattice

Caveats (II) Even e + e - or pp data can be fitted well! See, e.g., Becattini&Heinz(’97) What is the meaning of fitting parameters? See, e.g., Rischke(’02),Koch(’03) Why so close to T c ?  No chemical eq. in hadron phase!?  Essentially dynamical problem! Expansion rate  Scattering rate see, e.g., U.Heinz, nucl-th/

Matter in (Kinetic) Equilibrium? uu Kinetically equilibrated matter at rest Kinetically equilibrated matter at finite velocity pxpx pypy pxpx pypy Isotropic distribution Lorentz-boosted distribution

Radial Flow Blast wave model (thermal+boost) Kinetic equilibrium inside matter e.g. Sollfrank et al.(’93) Pressure gradient  Driving force of flow  Flow vector points to radial direction

Spectral change is seen in AA! Power law in pp & dAu Convex to Power law in Au+Au “Consistent” with thermal + boost picture Large pressure could be built up in AA collisions Adopted from O.Barannikova, (QM05)

Caveats (III) Flow reaches 50-60% of speed of light!? Radial flow even in pp? How does freezeout happen dynamically? STAR, white paper(’05)

Basic Checks  Necessary Conditions to Study the QGP at RHIC Energy density can be well above e c. –Thermalized? “Temperature” can be extracted. –Why freezeout happens so close to T c ? High pressure can be built up. –Completely equilibrated? Importance of systematic study based on dynamical framework

PART 3 (APPLICATIONS) Elliptic flow

Anisotropic Transverse Flow z x Reaction Plane x y Transverse Plane (perpendicular to collision axis)  Poskanzer & Voloshin (’98)

Directed and Elliptic Flow The 1 st mode, v 1 directed flow coefficient The 2 nd mode, v 2 elliptic flow coefficient x z Important in low energy collisions Vanish at midrapidity Important in high energy collisions x y

Ollitrault (’92) Hydro behavior Spatial Anisotropy Momentum Anisotropy INPUT OUTPUT Interaction among produced particles dN/d  No secondary interaction 0 22 dN/d  0 22 2v22v2 x y  What is Elliptic Flow? --How does the system respond to spatial anisotropy?--

Eccentricity: Spatial Anisotropy In hydrodynamics, :Energy density :Entropy density or x y

Eccentricity Fluctuation Interaction points of participants vary event by event.  Apparent reaction plane also varies.  The effect is relatively large for smaller system such as Cu+Cu collisions Adopted from D.Hofman(PHOBOS), talk at QM2006 A sample event from Monte Carlo Glauber model

Elliptic Flow in Hydro Saturate in first several femto-meters v 2 signal is sensitive to initial stage. Response of the system (= v 2 /) is almost constant. Pocket formula:v 2 ~0.2 Kolb and Heinz (’03)

Elliptic Flow in Kinetic Theory b = 7.5fm generated through secondary collisions saturated in the early stage sensitive to cross section (~1/m.f.p.~1/viscosity) v 2 is Zhang et al.(’99)ideal hydro limit t(fm/c) v2v2 : Ideal hydro : strongly interacting system

PART 3 (APPLICATIONS) Ideal hydrodynamic model

Inputs for Hydrodynamic Simulations for perfect fluids Final stage: Free streaming particles  Need decoupling prescription Intermediate stage: Hydrodynamics can be valid as far as local thermalization is achieved.  Need EOS P(e,n) Initial stage: Particle production, pre-thermalization?  Instead, initial conditions for hydro simulations 0 z t

Main Ingredient: Equation of State Latent heat Note: Chemically frozen hadronic EOS is needed to reproduce heavy particle yields. (Hirano, Teaney, Kolb, Grassi,…) Typical EOS in hydro models p=e/3 P.Kolb and U.Heinz(’03) EOS I Ideal massless free gas EOS H Hadron resonance gas EOS Q QGP: P=(e-4B)/3 Hadron: Resonance gas

Interface 1: Initial Condition Initial conditions (tuned to reproduce dN ch /d): initial time, energy density, flow velocity Transverse plane Reaction plane Energy density distribution (Lorentz-contracted) nuclei

Two Hydro Initial Conditions Which Clear the “First Hurdle” 1.Glauber model N part :N coll = 85%:15% 2. CGC model Matching I.C. via e(x,y, s ) Centrality dependenceRapidity dependence Kharzeev, Levin, and Nardi Implemented in hydro by TH and Nara

Interface 2: Freezeout --How to Convert Bulk to Particles-- Cooper-Frye formula Outputs from hydro in F.O. hypersurface  Contribution from resonance decays can be treated with additional decay kinematics.

Utilization of Hadron Transport Model for Freezeout (1) Sudden freezeout: QGP+hadron fluids (2) Gradual freezeout: QGP fluid + hadron gas Automatically describe chemical and thermal freezeouts 0 z t 0 z t At T=T f, =0 (ideal fluid) =infinity (free stream) T=TfT=Tf QGP fluid Hadron fluid QGP fluid

PART 3 (APPLICATIONS) Application of ideal hydrodynamic model to H.I.C. and comparison with data

Discovery of Perfect Fluidity!? Response=(output)/(input) Figures taken from STAR white paper(’05) Fine structure of elliptic flow Data reaches hydro limit curve QGP+hadron fluids with Glauber I.C.

Mug at RHIC/AGS User’s Meeting 2005 “RHIC serves the perfect liquid” A present from Raju when I gave a seminar talk at BNL.

Centrality Dependence of v 2 v 2 data are comparable with hydro results. Hadronic cascade cannot reproduce data. Note that, in v 2 data, there exists eccentricity fluctuation which is not considered in model calculations. hadronic cascade result (Courtesy of M.Isse) TH et al. (’06) QGP+hadron fluids with Glauber I.C.

Pseudorapidity Dependence of v 2 =0 >0<0 v 2 data are comparable with hydro results again around =0 Not a QGP gas  sQGP Nevertheless, large discrepancy in forward/backward rapidity TH(’02); TH and K.Tsuda(’02); TH et al. (’06). QGP only QGP+hadron QGP+hadron fluids with Glauber I.C.

Importance of Hadronic “ Corona ” Boltzmann Eq. for hadrons instead of hydrodynamics Including effective viscosity through finite mean free path QGP only QGP+hadron fluids QGP fluid+hadron gas T.Hirano et al.,Phys.Lett.B636(2006)299. QGP fluid+hadron gas with Glauber I.C.

More from Hybrid Model Mass dependence is o.k. Note: First result was obtained by Teaney et al % Centrality dependence is ok Large reduction from pure hydro in small multiplicity events T.Hirano et al.(’07) QGP fluid+hadron gas with Glauber I.C.

More from Hybrid Model(contd.) Adopted from S.J.Sanders (BRAHMS) QM2006 hybrid model AMPT QGP fluid+hadron gas with Glauber I.C.

Mass Ordering for v 2 (p T ) Mass dependence is o.k. from hydro+cascade % Proton Pion Mass ordering comes from rescattering effect. Interplay btw. radial and elliptic flows QGP fluid+hadron gas with Glauber I.C.

Summary So Far Glauber type initial condition followed by perfect fluid QGP + dissipative hadronic gas evolution is the most successful combination of the hydrodynamic modeling in description of flow data. What happens if one component is replaced? Especially, do the elliptic flow data admit viscosity in QGP?

v 2 (N part ) Depends on Initialization Glauber: Early thermalization Discovery of Perfect Fluid QGP CGC: No perfect fluid? Additional viscosity required in QGP Important to understand initial conditions much better for making a conclusion TH et al.(’06) Adil, Gyulassy, Hirano(’06) QGP fluid+hadron gas with CGC I.C.

Eccentricity from CGC Initial Condition x y Hirano and Nara(’04), Hirano et al.(’06) Kuhlman et al.(’06), Drescher et al.(’06)

Worthwhile to Remember pTpT v 2 (p T ) v2v2 If v 2 (p T ) = cp T, this holds exactly. This approximates v 2 (p T ) for pions. Near p T ~0, it should not be linear. v 2 eventually saturates since v 2 <1 0

Energy Dependence Integrated elliptic flow coefficient in mid-central collisions Differential elliptic flow coefficient in mid-central collisions STAR, white paper(’05)

Rapidity Dependence BRAHMS Integrated elliptic flow coefficient Differential elliptic flow coefficient PHOBOS 013013 Differential elliptic flow parameter might be tricky…

Summary (part 3) Hydrodynamic description is based on –local thermalization ansatz. All hydrodynamic results depend on –initial conditions –equation of states, and –final decoupling prescription Much attention should be paid to draw robust conclusion from hydrodynamic analyses. One should check what happens if the above conditions are changed.

Summary (part 3) (contd.) Elliptic flow has played a major role in understanding transport properties of the QGP. Glauber initial conditions, ideal hydro for the QGP, and dissipative gas for hadrons are three pillars for agreement. If CGC initial conditions are employed, one may need viscosity even in the QGP phase.

Hydro Talks at QM2008 Viscous Fluid Dynamics: –(Feb.6 th, Plenary) A.K.Chaudhuri; (Feb.8 th ) D.Molnar, R.Bhalerao, H.Song, A.Muronga; (Feb.9 th ) T.Kodama, P.Bozek Ideal Fluid Dynamics (incl. hybrid model) –(Feb.5 th ) M.Chojnacki; (Feb.8 th ) C.Nonaka, B.Betz, T.Hirano, T.Csorgo Viscosity –(Feb.5 th ) S.Gavin; (Feb.9 th ) K.Itakura

Flow Talks in QM2008 Plenary: S.Voloshin (anisotropic flow), N.Xu (collective flow) Parallel: Y.Lu (strangeness v 2, size dep.), P.Sorensen (v 2, fluctuation), M.Issah (size & energy dep. v 2 ), R.Raniwala (photon), S.Huang (high p T, v 2, v 4 ), R.Averback (single e), K.Miki (photon), C.Silvestre (J/psi), S.Raniwala(LHC)

References Far from Complete List … General –L.D.Landau, E.M.Lifshitz, Fluid Mechanics, Section –L.P.Csernai, Introduction to Relativistic Heavy Ion Collisions –D.H.Rischke, nucl-th/ –J.-Y.Ollitrault, [nucl-th]. Viscous hydro (general) –C.Eckart, Phys.Rev.15,919(1940). –W.Israel. J.M.Stewart, Ann.Phys.118,341(1979) –R.Maartens, astro-ph/ –I.Muller, Liv.Rev.Rel Recent reviews (including proceedings) –P.Kolb, U.Heinz, nucl-th/ –P.Huovinen,V.Ruuskanen, nucl-th/ –T.Hirano, [nucl-th]. –Y.Hama et al.,hep-ph/ –F.Grassi, nucl-th/ –C.Nonaka, nucl-th/

Thank You I would like to thank the organizers for giving me an opportunity to give a hydro lecture. I hope you enjoy QM2008 as well as your research life with hydrodynamics.

QGP mixed hadron Anisotropy of energy density distribution  Anisotropy of “Momentum” distribution TH&Gyulassy(’06) Time Evolution of a QGP Fluid

Digression (Dynamical) Viscosity : ~1.0x10 -3 [Pa s] (Water 20 ℃ ) ~1.8x10 -5 [Pa s] (Air 20 ℃ ) Kinetic Viscosity =/: ~1.0x10 -6 [m 2 /s] (Water 20 ℃ ) ~1.5x10 -5 [m 2 /s] (Air 20 ℃ ) [Pa] = [N/m 2 ] Non-relativistic Navier-Stokes eq.  water >  air BUT water < air

CAVEATS (III) Not necessary to be thermalized completely –Results from hadronic cascade models. How is radial flow generated dynamically? Finite radial flow even in pp collisions? –(T,v T )~(140MeV,0.2) –Is blast wave reliable quantitatively? Consistency? –Chi square minimum located a different point for  and  Flow profile? Freezeout hypersurface? Sudden freezeout?

p T Spectra from Glauber + QGP Fluid + Hadron Gas Model A hybrid model works well up to p T ~1.5GeV/c. Other components (reco/frag) would appear above.

Eccentricity from Universal Saturation Scale Almost no difference btw. conventional and universal definition for Qs^2 Conventional saturation scale Universal saturation scale near the origin 

v 2 (p T ) and v 2 (eta) from CGC initial conditions v 2 (model) > v 2 (data) 20-30%

Proof: Assuming,

Utilization of Hydro Results Jet quenching J/psi suppression Meson Recombination Coalescence Thermal radiation (photon/dilepton) Information along a path Information on surface Information inside medium Baryon J/psi c c bar

Jet Propagation through a QGP Fluid Color: parton density Plot: mini-jets Au+Au 200AGeV, b=8 fm transverse Fragmentation switched off hydro+jet model x y Full 3D ideal hydrodynamics + PYTHIA Parton distribution fn. pQCD 2  2 processes Fragmentation Gyulassy-Levai-Vitev formula Inelastic energy loss TH and Y.Nara (’02-)

Hydro+J/psi Model Gunji,Hamagaki,Hatsuda,TH(’07)